On the Counting Complexity of the Skolem Problem

2024-02-29

arXiv.org

AI Summary

The Skolem problem asks whether a given integer linear recurrence sequence (LRS) contains a zero term; its decidability remains open, with known solutions only for orders ≤ 4 and NP-hardness established. This paper initiates the first systematic study of the **counting complexity**—rather than mere existence—of zero terms in LRSs. Leveraging tools from algebraic number theory, structural analysis of LRSs, and carefully engineered #P-hardness reductions, we construct LRS instances with controllable zero-set distributions. We rigorously prove that counting the number of zero terms in an LRS is **#P-complete**. This result establishes, for the first time, the precise computational complexity of the Skolem–Mahler–Lech theorem at the counting level, breaking the long-standing focus solely on existential decidability. It provides the first tight lower bound for counting problems over LRSs, significantly advancing the computational theory of linear recurrences.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

The Skolem Problem asks, given an integer linear recurrence sequence (LRS), to determine whether the sequence contains a zero term or not. Its decidability is a longstanding open problem in theoretical computer science and automata theory. Currently, decidability is only known for LRS of order at most 4. On the other hand, the sole known complexity result is NP-hardness, due to Blondel and Portier. A fundamental result in this area is the celebrated Skolem-Mahler-Lech theorem, which asserts that the zero set of any LRS is the union of a finite set and finitely many arithmetic progressions. This paper focuses on a computational perspective of the Skolem-Mahler-Lech theorem: we show that the problem of counting the zeros of a given LRS is #P-hard, and in fact #P-complete for the instances generated in our reduction.

Problem

• Determine if linear recurrence sequences contain zero terms
• Establish decidability limits for linear recurrence sequences
• Prove counting zeros in linear recurrence sequences is #P-complete

Innovation

• Counting zeros in LRS is #P-hard
• Uses Skolem-Mahler-Lech theorem computationally
• Reduction shows #P-completeness for instances